Abstract. The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions. IntroductionCalculations of Feynman integrals arising in perturbative quantum field theory [4,5] reveal interesting patterns of zeta and multiple zeta values. Clearly, these are motivic in origin, arising from the existence of Tate mixed Hodge structures with periods given by Feynman integrals. We are far from a detailed understanding of this phenomenon. An analysis of the problem leads via the technique of Feynman parameters [12] to the study of motives associated to graph polynomials. By the seminal work of Belkale and Brosnan [3], these motives are known to be quite general, so the question becomes under what conditions on the graph does one find mixed Tate Hodge structures and multiple zeta values.The purpose of this paper is to give an expository account of some general mathematical aspects of these "Feynman motives" and to work out in detail the special case of wheel and spoke graphs. We consider only scalar field theory, and we focus on primitively divergent graphs. (A connected graph Γ is primitively divergent if #Edge(Γ) = 2h 1 (Γ) where h 1 is the Betti number of the graph; and if further for any connected proper subgraph the number of edges is strictly greater than twice the first Betti number.) From a motivic point of view, these play the role of "Calabi-Yau" objects in the sense that they have unique periods. Physically, the corresponding periods are renormalization scheme independent.Date: Sept. 28b, 2005. Graph polynomials are introduced in sections 1 and 2 as special cases of discriminant polynomials associated to configurations. They are homogeneous polynomials written in a preferred coordinate system with variables corresponding to edges of the graph. The corresponding hypersurfaces in projective space are graph hypersurfaces. Section 3 studies coordinate linear spaces contained in the graph hypersurface. The normal cones to these linear spaces are linked to graph polynomials of sub and quotient graphs. Motivically, the chain of integration for our period meets the graph hypersurface along these linear spaces, so the combinatorics of their blowups is important. (It is curious that arithmetically interesting periods seem to arise frequently (cf. multiple zeta values [11] or the study of periods associated to Mahler measure in the non-expansive case [8]) in situations where the polar locus of the integrand meets the chain of integration in combinatorially interesting ways.) Section 4 is not used in the sequel. It exhibits a natural resolution of singularities P(N) → X for a graph hypersurface X. P(N) is a projective bundle over projective space, and the fibres P(N)/X are projecti...

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For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in [Yu14] a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential d+df , extending a definition of Deligne in the case of curves. In this article, we show the degeneration at E 1 of the spectral sequence attached to the irregular Hodge filtration, by using the method of [Sab10]. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on his proof of the corresponding E 1 -degeneration, by reduction to characteristic p, when the pole divisor of the function is reduced with normal crossings. In Appendix E, M. Saito gives a different proof of the latter statement with a possibly non reduced normal crossing pole divisor.

The numerical characterization of bad linear subspaces claimed in the proposition of Section 3 is wrong; namely the implication d) ~ b) does not hold true.The following counterexample is due to A. Varchenko. Consider the direct product of two affine arrangements, each of them consisting of n >2 lines on a plane intersecting at one point; then take the projective closure of this arrangement. In this arrangement the singular point satisfies a) but does not satisfy c).The mistake occurs in the first line of p. 561, where we claim that the arrangement H'/= H'ilH'io again satisfies the assumption b), overlooking the possibility that H~'= H] for i+j. Therefore one can not argue inductively, the Euler characteristic being a topological invariant not taking in account the multiplicities. The easy implications a)~--~b) and c)~ b) remain true. The proof of the theorem (and of its corollary) does not use the numerical characterizations c), d) of (Bad) and therefore is not touched.We are grateful to A. Varchenko for drawing our attention to the above example.

Classically the vanishing of cohomology groups of a compact complex K~ihler manifold X with values in certain locally free sheaves ~#/is proved by studying positivity properties of the curvature form of a differentiable connection on ~/ compatible with the complex structure of X (e.g. [7]). If the Chern classes of J// are non trivial, the connection is neither holomorphic nor integrable. Therefore, trying to replace the differentiable connection with non trivial curvature by an integrable holomorphic connection V, one has at least to allow 17 to have poles along a "boundary divisor" D. We even will assume D to be a normal crossing divisor and 17 to have at most logarithmic poles along D. Since 17 is non singular and integrable on U--X -D, the positivity properties have to be replaced by topological properties of U together with conditions on the boundary behaviour of (Jt, 17).Because of the restriction made on the poles of 17 along D one has at disposal the theory of P. Deligne on differential equations with regular singular points [3] and in fact his Lecture Notes was the main source of inspiration of our work:Let V be the local constant system on U defined by sections of Jr'Iv, flat with respect to V and j: Ur In general it is quite difficult to decide when the spectral sequence degenerates (see (2.6)). Using Deligne's theory of mixed Hodge structures [4] Grauert-Riemenschneider's vanishing theorem (see (2.14,a)) as well as its generalization due to Y. Kawamata and the second author ((2.12) and (2.13)).If one drops the assumption on U, the degeneration of El(J/) implies the vanishing of certain natural restriction maps in cohomology. Applied to the sheaves ~-q~-I considered above one obtains the vanishing of the restriction maps of twisted differential forms in the cohomology ((3.2) and (3.3)). Especially one gets an improvement of the Koll~r-Tankeev vanishing theorems (3.5) as a direct interpretation of the degeneration of the spectral sequence.In w 1 we recall properties of sheaves with logarithmic connections and their De Rham complexes. The condition that the monodromies of V do not have 1 as eigenvalue implies that the minimal and the maximal extensions of V coincide, as we prove in (1.6).In w we give the cohomological interpretation of (1.6), provided the spectral sequence El(Jr' ) degenerates and U=X-D is affine. We discuss examples where all three assumptions hold and state and prove the vanishing theorems mentioned. w contains the applications to the cohomology of restriction maps, useful if U is not affine. The main observation is that the conditions posed on the monodromy of V imply that the residue maps obtained from V are surjective on each component of D and can be identified with the natural restriction map.In order to recover the positivity properties of differentiable connections in terms of the logarithmic connection 17, one should at least be able to define the Chern classes in the De Rham cohomology. That is done in Appendix B by describing the Atiyah class as the image of the residue of 17,...